theorem Th50:
  a <> 0 & (a * v) + W = the carrier of W implies v in W
proof
  assume that
A1: a <> 0 and
A2: (a * v) + W = the carrier of W;
  assume not v in W;
  then not 1 * v in W by RLVECT_1:def 8;
  then not (a" * a) * v in W by A1,XCMPLX_0:def 7;
  then not a" * (a * v) in W by RLVECT_1:def 7;
  then
A3: not a * v in W by Th21;
  0.V in W & a * v + 0.V = a * v by Th17;
  then a * v in {a * v + u : u in W};
  hence contradiction by A2,A3;
end;
